5 Matter Waves Chapter Outline 5.1 The Pilot Waves of de Broglie 5.7 The Wave–Particle Duality De Broglie’s Explanation of The Description of Electron Diffraction Quantization in the in Terms of ⌿ Bohr Model A Thought Experiment: Measuring 5.2 The Davisson–Germer Experiment Through Which Slit the Electron The Electron Microscope Passes 5.3 Wave Groups and Dispersion 5.8 A Final Note Matter Wave Packets Summary 5.4 Fourier Integrals (Optional) Constructing Moving Wave Packets 5.5 The Heisenberg Uncertainty Principle A Different View of the Uncertainty Principle 5.6 If Electrons Are Waves, What’s Waving? In the previous chapter we discussed some important discoveries and theo- retical concepts concerning the particle nature of matter. We now point out some of the shortcomings of these theories and introduce the fascinating and bizarre wave properties of particles. Especially notable are Count Louis de Broglie’s remarkable ideas about how to represent electrons (and other particles) as waves and the experimental confirmation of de Broglie’s hypothesis by the electron diffraction experiments of Davisson and Germer. We shall also see how the notion of representing a particle as a localized wave or wave group leads naturally to limitations on simultaneously mea- suring position and momentum of the particle. Finally, we discuss the passage of electrons through a double slit as a way of “understanding” the wave–particle duality of matter. 151 Copyright 2005 Thomson Learning, Inc. All Rights Reserved. 152 CHAPTER 5 MATTER WAVES 5.1 THE PILOT WAVES OF DE BROGLIE By the early 1920s scientists recognized that the Bohr theory contained many inadequacies: • It failed to predict the observed intensities of spectral lines. • It had only limited success in predicting emission and absorption wave- lengths for multielectron atoms. • It failed to provide an equation of motion governing the time develop- ment of atomic systems starting from some initial state. • It overemphasized the particle nature of matter and could not explain the newly discovered wave–particle duality of light. • It did not supply a general scheme for “quantizing” other systems, espe- cially those without periodic motion. The first bold step toward a new mechanics of atomic systems was taken by Louis Victor de Broglie in 1923 (Fig. 5.1). In his doctoral dissertation he pos- tulated that because photons have wave and particle characteristics, perhaps all forms of matter have wave as well as particle properties. This was a radical idea with no Figure 5.1 Louis de Broglie experimental confirmation at that time. According to de Broglie, electrons was a member of an aristocratic had a dual particle–wave nature. Accompanying every electron was a wave French family that produced (not an electromagnetic wave!), which guided, or “piloted,” the electron marshals, ambassadors, foreign through space. He explained the source of this assertion in his 1929 Nobel ministers, and at least one duke, prize acceptance speech: his older brother Maurice de On the one hand the quantum theory of light cannot be considered satisfactory Broglie. Louis de Broglie came since it defines the energy of a light corpuscle by the equation E ϭ hf containing the rather late to theoretical physics, frequency f. Now a purely corpuscular theory contains nothing that enables us to as he first studied history. Only define a frequency; for this reason alone, therefore, we are compelled, in the case of after serving as a radio operator light, to introduce the idea of a corpuscle and that of periodicity simultaneously. On in World War I did he follow the the other hand, determination of the stable motion of electrons in the atom intro- lead of his older brother and duces integers, and up to this point the only phenomena involving integers in begin his studies of physics. physics were those of interference and of normal modes of vibration. This fact sug- Maurice de Broglie was an out- gested to me the idea that electrons too could not be considered simply as corpus- standing experimental physicist cles, but that periodicity must be assigned to them also. in his own right and conducted experiments in the palatial fam- Let us look at de Broglie’s ideas in more detail. He concluded that the ily mansion in Paris. (AIP Meggers wavelength and frequency of a matter wave associated with any moving object Gallery of Nobel Laureates) were given by h De Broglie wavelength ␭ ϭ (5.1) p and E f ϭ (5.2) h where h is Planck’s constant, p is the relativistic momentum, and E is the total rel- ativistic energy of the object. Recall from Chapter 2 that p and E can be written as p ϭ ␥mv (5.3) and E 2 ϭ p 2c 2 ϩ m2c4 ϭ ␥2m2c4 (5.4) Copyright 2005 Thomson Learning, Inc. All Rights Reserved. 5.1 THE PILOT WAVES OF DE BROGLIE 153 where ␥ ϭ (1 Ϫ v 2/c 2)Ϫ1/2 and v is the object’s speed. Equations 5.1 and 5.2 immediately suggest that it should be easy to calculate the speed of a de Broglie wave from the product ␭f. However, as we will show later, this is not the speed of the particle. Since the correct calculation is a bit complicated, we postpone it to Section 5.3. Before taking up the question of the speed of matter waves, we prefer first to give some introductory examples of the use of ␭ ϭ h/p and a brief description of how de Broglie waves provide a physical picture of the Bohr theory of atoms. De Broglie’s Explanation of Quantization in the Bohr Model Bohr’s model of the atom had many shortcomings and problems. For exam- ple, as the electrons revolve around the nucleus, how can one understand the fact that only certain electronic energies are allowed? Why do all atoms of a given element have precisely the same physical properties regardless of the infinite variety of starting velocities and positions of the electrons in each atom? De Broglie’s great insight was to recognize that although these are deep problems for particle theories, wave theories of matter handle these problems neatly by means of interference. For example, a plucked guitar string, although initially subjected to a wide range of wavelengths, supports only r standing wave patterns that have nodes at each end. Thus only a discrete set of wavelengths is allowed for standing waves, while other wavelengths not included in this discrete set rapidly vanish by destructive interference. This same reasoning can be applied to electron matter waves bent into a circle around the nucleus. Although initially a continuous distribution of wave- lengths may be present, corresponding to a distribution of initial electron λ velocities, most wavelengths and velocities rapidly die off. The residual stand- ing wave patterns thus account for the identical nature of all atoms of a given Figure 5.2 Standing waves fit element and show that atoms are more like vibrating drum heads with discrete to a circular Bohr orbit. In this modes of vibration than like miniature solar systems. This point of view is particular diagram, three wave- emphasized in Figure 5.2, which shows the standing wave pattern of the lengths are fit to the orbit, cor- electron in the hydrogen atom corresponding to the n ϭ 3 state of the Bohr responding to the n ϭ 3 energy theory. state of the Bohr theory. Another aspect of the Bohr theory that is also easier to visualize physically by using de Broglie’s hypothesis is the quantization of angular momentum. One simply assumes that the allowed Bohr orbits arise because the elec- tron matter waves interfere constructively when an integral number of wavelengths exactly fits into the circumference of a circular orbit. Thus n␭ ϭ 2␲r (5.5) where r is the radius of the orbit. From Equation 5.1, we see that ␭ ϭ h/mev. Substituting this into Equation 5.5, and solving for mevr, the angular momen- tum of the electron, gives m evr ϭ nប (5.6) Note that this is precisely the Bohr condition for the quantization of angular momentum. Copyright 2005 Thomson Learning, Inc. All Rights Reserved. 154 CHAPTER 5 MATTER WAVES EXAMPLE 5.1 Why Don’t We See the Wave Properties of a Baseball? An object will appear “wavelike” if it exhibits interference h h ␭ ϭ ϭ or diffraction, both of which require scattering objects or p √2mqV apertures of about the same size as the wavelength. A baseball of mass 140 g traveling at a speed of 60 mi/h (b) Calculate ␭ if the particle is an electron and (27 m/s) has a de Broglie wavelength given by V ϭ 50 V. h 6.63 ϫ 10Ϫ34 Jиs Solution The de Broglie wavelength of an electron ␭ ϭ ϭ ϭ 1.7 ϫ 10Ϫ34 m p (0.14 kg)(27 m/s) accelerated through 50 V is Ϫ15 h Even a nucleus (whose size is Ϸ 10 m) is much too ␭ ϭ large to diffract this incredibly small wavelength! This √2m eqV explains why all macroscopic objects appear particle-like. 6.63 ϫ 10Ϫ34 Jиs ϭ √2(9.11 ϫ 10Ϫ31 kg)(1.6 ϫ 10Ϫ19 C)(50 V) EXAMPLE 5.2 What Size “Particles” Do Ϫ10 Exhibit Diffraction? ϭ 1.7 ϫ 10 m ϭ 1.7 Å A particle of charge q and mass m is accelerated from This wavelength is of the order of atomic dimensions and rest through a small potential difference V. (a) Find its the spacing between atoms in a solid. Such low-energy de Broglie wavelength, assuming that the particle is non- electrons are routinely used in electron diffraction exper- relativistic.